Saturday, September 29, 2012

Cool Math - I

Current set of cool math problems (solutions later). See if you can see through them. Best of luck.

Note: I have solutions to a lot of these questions, almost all. But, it could be that some solution slipped by time, since I used these long time ago. In that case, I might not have some solutions.

*1. Consider a `4n` * `4n` square. Now, consider rectangles of integer dimensions which would fit inside this square (max dimension of them can be `4n`). What is the probability (considering very large `n`), that these rectangles have an area lesser than or equal to `4n^{2}` chosen over all possible dimensions (that would fit inside the square).
*2. Prove that, if a number `n` has the prime number expansion form of `p_{1}^{k_{1}}*p_{2}^{k_{2}}*p_{3}^{k_{3}}...`, the sum of its factors is (where `1` and `n` are also considered factors) is given by `(p_{1}^{k_{1}+1} - 1) / (p_{1} - 1) * (p_{2}^{k_{2}+1} - 1) /(p_{2} - 1) * ...`
*3. From the above definition of factors, prove that a perfect number has sum of its factors, equal to twice the number.
**4. Prove fermat's little theorem: If `p` is a prime, `a^{p} \equiv a (mod p)`. (Hint: Use binomial expansion to power p, and mathematical induction on a).
5. Prove that if `p` is not a prime, then `2^{p} - 1` is not a prime.
**6. Prove that all even perfect numbers are of the form, `2^{p−1} * (2^{p}−1)` where `(2^{p}−1)` is a prime.
**7. Prove that if both `p` and `(2p + 1)` are primes, `(2p + 1)` will be a factor of `(2^{p}−1)`.
8. A triangular number (`n` th) is a number, which is got by summing the first `n`, positive integers. Prove that every number of the form `2^{p−1} * (2^{p}−1)`, is a triangular number.
9. Derive a formula for 'sum of first `n` odd cubes', and prove that every number of the form `2^{p−1} * (2^{p}−1)`, is a sum of first `n` odd cubes.
10. Even perfect numbers (except 6) give a remainder of 1, when divided with 9.
11. Subtracting 1 from a perfect number, and dividing it by 9, always gives a perfect number.
12. Summing all digits in a number, and summing all the digits in the resulting number, and repeating the process till a single digit is obtained is called, getting the "digital root" of a number. Prove that a positive number is divisible by 9, iff, its digital root is 9.
13. Any number of the form `2^{m−1} * (2^{m}−1)`, where m is any odd integer, leaves a remainder of 1 when divided with 9.
*14. An Ore's harmonic number, is a number whose factors have a harmonic mean of an integer. Prove that every perfect number is an Ore's harmonic number.
*15. Prove that, for any perfect number, sum of reciprocals of its factors is equals to two.
*16. Prove that for any integer `M`, the product of arithmetic mean of its factors and harmonic mean of the factors, is the number `M` itself.
*17. Any odd perfect number `N` is of the form, `N = q^{\alpha} * p_{1}^{2e_{1}} * p_{2}^{2e_{2}} * p_{3}^{2e_{3}} ...p_{k}^{2e_{k}}`. Where `q`, `p_{1}`, `p_{2}` are all distinct primes, with `q \equiv \alpha \equiv 1 (mod 4)`.
18. Continuing on 17, the smallest prime factor of `N` is less than `(2k + 8) / 3`.
19. Continuing on 18, prove also that `N < 2^{4^{k+1}}`.
20. Continuing on 19, prove also that the largest prime factor of `N` is greater than `10^{8}`.
21. Continuing on 20, prove also that `q^{\alpha} > 10^{62}` or `p_{j}^{2e_{j}} > 10^{62}` for some `j`.
22. Continuing on 21, prove also that the largest prime factor of `N` is greater than `10^{8}`.
23. Continuing on 22, prove also that the second largest prime factor is greater than `10^{4}`, and the third largest prime factor is greater than 100.
24. Continuing on 23, prove also that if N has at least 101 prime factors and at least 9 distinct prime factors. If 3 is not one of the factors of N, then N has at least 12 distinct prime factors.
25. Prove that an odd perfect number is not divisible by 105.
26. Every odd perfect number is of the form N ≡ 1 (mod 12), N ≡ 117 (mod 468), or N ≡ 81 (mod 324).
27. The only even perfect number of the form x3 + 1 is 28.
28. 28 is also the only even perfect number that is a sum of two positive integral cubes
29. Prove that no perfect number can be a square number.
30. The number of perfect numbers less than `n` is less than `c\sqrt{n}`, where `c > 0` is a constant.
31. Numbers where the sum is less than the number itself are called deficient, and where it is greater than the number, abundant. Give an example of each.
32. In number theory, a practical number or panarithmic number is a positive integer `n` such that all smaller positive integers can be represented as sums of distinct divisors of `n`. Prove that any even perfect number and any power of two is a practical number.
33. Prove that prime numbers are infinite. (Use proof by contradiction).
34. Prove that `2n_{C_{n}}` is divisible by a prime number n, iff, n's p-ary repesentation contains all digits less than `p / 2`.
35. Prove that if `N` is a perfect number, none of its multiples or factors is a perfect number.
36. when written in decimal notation, how many trailing zeros are present in 32!. Can you generalize it for `n`.
37. Prove that only square numbers have odd number of factors.
38. Prove that the number of factors of a number `n`, is always not greater than `2 * \lfloor\sqrt{n}\rfloor`.
39. Continuing on 38, is there any number for which the number of factors is equal to the upper bound? Why?
40. when written in decimal notation, what digit do powers of 5 end with? What about powers of 6?
41. Prove that in `x^{4} + y^{4} = z^{4}` (x, y and z are positive integers), at least one of x or y is divisible by 5.
42. Consider a number with maximum number of factors between 1 and `n^{2}`. Prove that such a number is not divisible by any prime number `p >= n`, for `n > 4`.
43. Prove that `n!` cant be a square number for any `n > 1`. (My proof involves the proven conjecture that there exists a prime number between `2n` and `3n`, for `n > 1`).
44. Every prime number `p > 3` satisfies the following: `p^{2} = 24*k + 1` for some positive number `k`.
45. Also, every prime number `p > 30` satisfies the following: `p^{4} = 240*k + 1` for some positive number `k`.
46. Prove that a composite number (`N`) definitely has a prime factor (`<= \lfloor\sqrt{n}\rfloor`).
47. Prove that the fraction of "the number of divisors of first n prime numbers", in any `N` numbers is same as `(1 - 1/2) * (1 - 1/3) * (1 - 1/5) * ...` (Denominators are all first `n` prime numbers. Hint: Generalize Eratosthenes Sieve).
48. Using `+`, `-`, `*`, `/`, `\sqrt`, `.` (decimal), `!` (factorial) and `^` (to the power of), find a way to represent 73 using four fours.
49. Do same as 48, for 77.
50. Do same as 48, for 87.
51. Do same as 48, for 93.
52. Do same as 48, for 99.
53. There is a way to represent all positive numbers using repeated trigonometric functional application on a single 4. Could you get it?
54. Prove that successive terms of a fibonacci sequence are relatively prime.
55. Prove that if the `n` th fibonacci number is denoted as `F_{n}`, `F_{n} | F_{mn}` for any m, n.
56. Consider the fibonacci sequence modulo some positive number `k`. Prove that modulo `k`, the fibonacci sequence has zeros equally spaced (periodically arranged).
57. Prove that the `n` th prime number, is less than `n^{2}`, for `n > 1`.
58. Prove that, in every 6 numbers (numbers themselves greater than 6), there can only be 2 prime numbers.
59. Continuing 58, also prove that every 15 numbers, there can only be 4 primes.
60. Continuing 59, also prove that every 35 numbers, only 8 can be primes.
61. Continuing 60, can you generalize it.
62. Continuing 61, can you generalize it as a formula representing number of prime numbers from 1 to `N`.
63. Continuing 62, we seem to sort of know when the number of primes exceeds the known prime numbers so far. So, does it give a way of finding where next prime would lie? (Well, it did not give a precise enough formula for me. Looked very promising though.)
64. Prove that the fraction of numbers relatively prime to 3 are `2/3`. Prove that the fraction of numbers relatively prime to 5 are `4/5`. Also, prove that the fraction of numbers which are relatively prime to `15` are `8/15`.
65. Continuing on 64, what do these results indicate?
66. Continuing on 65, can you generalize it (the fraction of relatively prime numbers to a number `n`) for an arbitrary number `n` (Consider the prime power expansion of `n`).
67. Show that if a number is deficient, all of its factors are deficient too.
68. Show that if a number is abundant, all of its multiples are abundant too.
69. Based on 67 and 68, devise a method to stop searching for perfect numbers, when starting search from 1 and going upwards.
70. Prove that no prime number is a perfect number.
71. Prove that no number of the form `p_{1}^{k_{1}}`, where `p_{1}` is a prime can be a perfect number.
72. Prove that no number of the form `p_{1}^{k_{1}} * p_{2}^{k_{2}}`, where `p_{1} > 2` and `p_{2} > p_{1}` are primes can be a perfect number.
73. Prove that, if all prime factors of a perfect number (`N`) are more than `p`, prove that `N` must have at least `log_{p/(p-1)}^{2}` number of prime factors.
74. Prove that, for any prime number p, `a^{(p-1)*n} + b^{(p-1)*n} = c^{(p-1)*n}` for any positive n, and positive numbers a, b and c, implies that at least one of `a`, `b` and `c` are divisible by `p`.
75. Prove that any prime factor of `(a^{5} - b^{5}) / (a - b)` is of the form of `10*k + 11`, for some non-negative k, or it is `5`.
76. Generalize 75, generalize about any prime factor of `(a^{p} - b^{p}) / (a - b)` where `p` is a prime number.
77. Continuing on 75, also prove that 5 is a factor of that expression, iff `(a - b)` is divisible by 5.
78. Generalize similarly for a generic prime exponent `p`.
79. Depending on 75-78, consider a equality of the form `a^{5} - b^{5} = c^{5} - d^{5}`. Prove that if `(a - b)` is divisible by a prime number `p` not of the form `10*k + 11`, then `(c - d)` also is divisible by `p`. (Taxicab problem related).
80. Can we say something about `a` (or `b`) in the equation, `a^{p} + b^{p} = c^{p}`. What observation can be made about it?
81. How many 2 digit numbers are divisible by the digits in them. (If same digit occurs multiple times, divide multiple times with the digit. Exclude zeroes from divisors)
82. How many three digit numbers are divisible by all digits in them
83. If `n! + 1 = m^{2}` has solutions, then `n!` (consider `n` beyond trivial ones for which we know some solution, like, 4, 5 and 7) can be factorized into two numbers `a` and `b` such that `a - b = 2`, and `ab = n!`.
84. Continuing on 83, `a` and `b` are such that, one of them is divisible by 2 but not by 4.
85. Continuing on 84, `a` and `b` are such that, every prime number `n > p > 2`, is a factor of exactly one of them, but not both.
86. Continuing on 85, `a` is of the form `2 * op_{1}^{k_{1}} * op_{2}^{k_{2}} * op_{3}^{k_{3}} * ...` and `b` is of the form `2^{k} * op_{a}^{k_{a}} * op_{b}^{k_{b}} * op_{c}^{k_{c}} * ...` where `op_{i}` are all distinct odd primes.
87. Continuing on 86, you have to prove that for large `k`, `op_{1}^{k_{1}} * op_{2}^{k_{2}} * op_{3}^{k_{3}} ... + 1` (or `-1`), where `op_{i}` are odd-primes cannot have high powers of 2 as a factor (for solving the Brocard's problem). Proof, anyone?
88. Continuing on 87, prove that `3^{n} + 1` is divisible by either 2 or 4, but not by higher powers of 2.
89. Continuing on 88, prove that `5^{n} + 1` is always divisible by 2, but not by higher powers of 2.
90. Continuing on 89, prove that `p^{2n} + 1`, where `p` is a prime number, is always divisible by 2, but not by 4.
91. Continuing on 90, prove that `p^{2n+1} + 1` is divisible by the same power of 2, that divides `p + 1`.
92. Continuing on 91, prove that `3^{2n+1} - 1` is always divisible by 2, but not by higher powers of 2.
93. Continuing on 92, prove that `5^{2n+1} - 1` is always divisible by 4, but not by higher powers of 2.
94. Continuing on 93, prove that `p^{2n + 1} - 1`, where `p` is a prime number, is always divisible by the same power of 2, that divides `p - 1`.
95. Continuing on 94, prove that `p^{2n} - 1`, where `p` is a prime number, is always divisible by same power of 2 (`r`), unless `n` is a power of 2 itself.
96. Continuing on 95, prove that `p^{2^{n}} - 1`, where `p` is a prime number, is always divisible by `2^{n+k}`, where `2^{k}` divides `p^{2i} - 1` maximally, where `i` is not of the form `2^{o}`.
97. Continuing on 96, one can give simpler formulation of problem in 83. For this, prove that `2^{n}` is never a factor of `n!`.
98. Continuing on 97, prove that `n!` always has a factor of the form `2^{(1 - \delta) * n}`, where `\delta` decreases with increasing `n`.
99. Continuing on 98 and 83, for such an `n` to exist, you have to prove that for large `n`, `op_{1}^{k_{1}} * op_{2}^{k_{2}} * op_{3}^{k_{3}} ... + 1` (or `-1`), where `op_{i}` are odd-primes should divide `2^{(7 * n)/8}` (`(7 * n) / 8` is for instance).
100. Prove that `2^{p−1} * (2^{p}−1)` is an even perfect number whenever `2^{p}−1` is a prime (Euclid).
101. Prove that, if `p` is an odd number such that `p + 1 = 2^{k_{1}}*o_{1}`, where `o_{1}` is an odd number and `k_{1} > 1`, and `q` is another similar number (like `p`), then `pq + 1` is divisible by 2 and not by 4.
102. Prove that, if `p` is an odd number such that `p + 1 = 2*o_{1}`, where `o_{1}` is an odd number, and `q` is another similar number (like `p`), then `pq + 1` is divisible by 2 and not by 4.
103. Prove that, if `p` is an odd number such that `p - 1 = 2^{k_{1}}*o_{1}`, where `o_{1}` is an odd number and `k_{1} >= 1`, and `q` is another similar number (like `p`), then `pq - 1` is divisible by `min(k_{1}, k_{2})` unless `k_{1} = k_{2}`.
104. Prove that, if `p` is an odd number such that `p + 1 = 2^{k_{1}}*o_{1}`, where `o_{1}` is an odd number, and `q` is another similar number (like `p`), then `pq + 1` is divisible by highers powers of 2 than (`2^{1}`), iff, exactly one of `k_{1}` and `k_{2}` is equal to `1`.
105. Prove that, `op_{1}^{k_{1}} * op_{2}^{k_{2}} * op_{3}^{k_{3}} * ... + 1` where `op_{i}` are odd primes, can be reduced to something that divides 2 but not 4, or, something of the form `(2*o_{1} - 1) * (2^{k}*o_{2} - 1) + 1` for some odd numbers `o_{1}` and `o_{2}`.
106. State and prove the conditions where the expression in 105 divides only 2 but not 4.
107. Consider the case in 105, where `o_{1} = 1`. In this case, prove that it does not result in an `n` such that `n! + 1 = m^{2}`. (`k` is not as big as `(7*n)/8`).
108. Continuing on 103, prove that all prime numbers between `n/i` and `n/(i+1)` (where `i >= 1`) have the same power in `n!`.
109. Continuing on 103, prove that only the prime numbers between `n/2` and `n` have an exponent of `1`, in the prime number expansion of `n`.
110. Continuing on 10, consider `(2^{k}*o_{2} - 1) = 3^{k_{1}}` (Since any such prime number would do). Prove that if `3^{k_{1}}` divides `2^{l} - 1`, then `l` is such that `l = (4 * k_{1} - 2)`.
111. Prove that, `\pi . p_{i}^(1 / (p_{i} - 1))` (Here, `\pi` means product of), where `p_{i}` are prime numbers (starting from `2`), does not converge as `i` increases.
112. Continuing on 111, prove that the lower-bound of the expression in 111, is, `(2\pi n)^{1/(2n)} . n/e` where `e` is the natural logarithm base (Hint: Remember something, yeah!).
113. Prove that, if `2n` numbers are present, where the minimum difference between then is `\delta` and maximum difference is a polynomial function of `\delta` with constant term zero, then any difference between two terms formed by multiplying `n` each numbers is always divisible by `\delta`.
114. Prove that the exponent of prime number `p` in `n!` is always not less than `(n + 1)/(p -1) * (1 - 1/(p^{\lfloor log_{p}^{n} \rfloor})) - \lfloor log_{p}^{n} \rfloor`.
115. Prove that the exponent of prime number `p` in `n!` is always not greater than `n/(p -1) * (1 - 1/(p^{\lfloor log_{p}^{n} \rfloor}))`
116. Prove that, all numbers between `n/i` and `n/(i + 1)` have same power in prime number expansion of `n!`, whenever `i < \sqrt (n)`.
117. Continuing on 116, prove that, for two prime numbers `p_{1}` and `p_{2}`, belonging to same interval (`p_{1} > p_{2}`), and having exponent `k` in prime number expansion of `n!`, the ratio of `p_{1}^{k}` to `p_{2}^{k}` is always less than `e` (the natural log base).
118. Also prove that, when a solution to Brocard's problem exists, the prime terms in expansion of `n!`, should be distributed across two terms (`A` and `B`) such that, `A - B = 1`, and `AB * 4 = n! + 1`.
119. Continuing on 118, prove that either `A` or `B` contains `2^({k_{1}} - 2)`, where `k_{1}` is the exponent of `2` in `n!` (meaning, its prime number expansion).
120. Continuing on 119, prove that if equal number of prime numbers are segregated into `A` and `B`, a solution to Brocard's problem is not possible. (Use 113).
121. Continuing on 117, if prime numbers within same interval, are factored such that equal number of them are in `A` and in `B`, then give an estimate of their ratio, in terms of `e` (Reduce the ratio as much as possible).
122. Prove that for large `n`, the smallest term, in the prime number expansion of `n!` is always not lesser than `(n + 1) / 2`.
123. MRB constant is given by , `\sigma_{n=1} (-1)^{n} * 1/(n^{n})` (till infinity). Prove that, for any first `n` terms of the MRB constant (denoted by `MRB(n)`), `(MRB(n))^{n!^{n}}` is a rational number.
124. Prove that, any sub-sequence of terms for the MRB constant, is not a transcendental number.